On perturbed metric-preserved mappings and their stability characterizations

Let X, Z be two real Banach spaces, and epsilon >= 0. In this paper, we first show that if there is a mapping f : X -> Z with f(0) = 0 satisfying vertical bar parallel to f(x) - f(Y)parallel to - f(y)parallel to - parallel to x - y parallel to vertical bar <= epsilon for all x,y is an element of X, then we can define a linear sun active isometry U : X*-> Z* / N for some closed subspace N of Z* by an in-variant mean of X. We then show that the subspace N plays a crucial role in this paper. For example, (1) U* :N-perpendicular to -> X** is a w*-to-w* continuous surjective isometry; and, in particular, if Y = span(over bar) f(X) is reflexive, then the mapping f is stable if and only if N-perpendicular to is complemented in Y; (2) if Y is reflexive and N-perpendicular to is complemented in Y, then for any projection P :Y -> N-perpendicular to, the operator T = U*P satisfies parallel to Tf(x) - x parallel to <= 4 epsilon, for all x is an element of X; and (3) if, in addition, Y is Gateaux smooth and locally uni-formly convex, then T = U* P satisfies the sharp estimate parallel to Tf(x) - x parallel to <= 2 epsilon, for all x is an element of X; We present similar results for such mappings on general Banach spaces. (C) 2014 Elsevier Inc. All rights reserved.